I took a break from playing Crusader Kings II for a few months. Since I've come back (November 2012), I'm suddenly unable to raise levies from most of my counties. The button is grayed out, even though there are some levies available to raise in the county. (The green/red bar for the levy in the county info screen is mostly green.) The most distressing part is that some of those counties could raise ships, and so now I have more troops than transport.

I tried raising all available troops from the Military screen, but most counties don't have any army on them. How did raising levies change?

In patch v1.07, you raise fewer, larger levies from your direct
vassals (we call these "Liege Levies" for lack of a better term.)
These levies are now handled separately from actual Holding levies;
you only get regular holding levies from your own Demesne. The
opinions of vassals of vassals are thus now mostly irrelevant; you
only need to worry about the opinions of your direct vassals. Another
benefit is that you don't get tiny little armies all over the place
when you mobilize your realm. Liege levies also fix a fundamental
balancing flaw in that you can no longer just keep raising little
levies from all vassals as a war drags on (due to the portion of
holding levies that was reserved for the liege.) The new Liege Levies
do not reinforce at home while raised.

in one of the latest updates (patch 1.07) they made it so instead of a whole lot of mini-levies, now you get one big levy per direct vassal. You only get regular holding levies from your own demesne possessions.

This sounds like you currently have levies raise from those holdings. You can always only raise one army from a holding (no matter how small or large the army is). And until that army is completely destroyed or until you disband it, you won't be able to raise any levies from that holding.